2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/21782We present a general technique to compute how the energy of a configuration varies as a function of its overlap with the ground state in the case of optimization problems. Our approach is based on a generalization of the cavity method to a system interacting with its ground state. With this technique we study the random matching problem as well as the mean field diluted spin glass. As a byproduct of this approach we calculate the de Almeida-Thouless transition line of the spin glass on a fixed connectivity random graph.13 pages, 7 figuresDisordered Systems and Neural NetworksStatistical Mechanicstext